Degenerate dispersive effects in partial and lattice differential equations

当一个波的传播速度取决于该波的频率时，一个空间扩展系统被称为色散系统。线性色散效应在许多物理场景的研究中起着重要的作用，关于半线性色散方程的结果已经大量出现。然而，在某些情况下，波的传播速度不仅取决于波的振幅，还取决于波的频率。也就是说产生色散的机制是非线性的。本项目的目的是为非线性色散效应退化（即，当振幅趋于零时，它们很快消失）时的这些方程发展数学上严格的理论。方程的简并性允许存在不光滑的经典解，并且有大量的数值和形式证据表明，在这种情况下，简并色散导致类似于反热方程的灾难性不稳定性。因此，该项目的主要目标之一是发展退化色散偏微分方程的存在性理论。简并色散方程作为硬球链和晶格的动力学模型而出现。实验结果表明，这种链具有紧密聚焦的脉冲，易于产生和高度可调谐。这些窄脉冲有望用于无损检测、减震、遥感和医学成像。目前，对于这些脉冲的稳定性和鲁棒性还没有理论上严格的解释。虽然也有关于类似物理问题的结果，其中色散效应不是简并的，但它们强烈依赖于慢运动脉冲非常宽且振幅非常小的事实。在硬球的情况下，有趣的脉冲解具有固定的宽度，与速度和振幅无关，如何将已知的结果推广到这种情况并不明显。该项目的一个主要目标是开发用于研究此类脉冲的稳定性和鲁棒性的数学工具。研究生和本科生的包容和培训是这个项目的一个组成部分。在其他项目中，本科生将参与研究交通流模型这一重要应用。

A spatially extended system is said to be dispersive when the speed of propagation of a wave depends upon that wave's frequency. Linear dispersive effects play a fundamental role in the study of a large number of physical scenarios and there has been an explosion of results concerning semi-linear dispersive equations. Nevertheless there are situations in which the speed of propagation of a wave depends on the wave's frequency as well as its amplitude. That is to say the mechanism which generates dispersion is nonlinear. It is the purpose of this project to develop mathematically rigorous theory for such equations when the nonlinear dispersive effects are degenerate (i.e., they vanish very rapidly as the amplitude tends to zero). The degeneracy of the equations allows for the existence of classical solutions which are not smooth and there is substantial numerical and formal evidence that cases arise in which degenerate dispersion leads to catastrophic instability akin to that of a backwards heat equation. Consequently, one of the principal goals of the project is to develop the existence theory for degenerate dispersive partial differential equations.Degenerate dispersive equations arise as models for the dynamics of chains and lattices of hard spheres. Experimental results demonstrate that such chains exhibit tightly focused pulses which are easily generated and highly tunable. These narrow pulses are expected to find uses in non-destructive testing, shock absorption, remote sensing and medical imaging. At this time, there is no theoretically rigorous explanation for the stability and robust nature of these pulses. Though there are results concerning similar physical problems where the dispersive effects are not degenerate, they rely strongly on the fact that slow moving pulses are very wide and very small in amplitude. In the case of hard spheres, the interesting pulse solutions have a fixed width, independent of speed and amplitude and it is not obvious how to extend the known results to this case. A primary goal of the project is developing mathematical tools for the study of the stability and robustness of such pulses. The inclusion and training of graduate and undergraduate students is an integral part of this project. Among other projects, undergraduate students will be involved into investigation of an important application, models for traffic flow.